18.969 | Fall 2006 | Graduate

Topics in Geometry: Dirac Geometry

Lecture Notes

The lecture notes were prepared by students in the class: Yanki Lekili, Jacob Bernstein, Chriss Kottke, Ana Rita Pires, James Pascaleff, Nick Rozenblyum, and Kartik Venkatram.

The complete set of lecture notes (PDF - 1.2 MB)

Lecture 1: Smooth manifolds, geometry of foliations, and symplectic structure. (PDF)

Lecture 2: Comments on previous lecture, symplectic manifolds, and Poisson geometry. (PDF)

Lecture 3: Almost complex structure, Hermitian structure, integrability of J, forms on a complex manifold, and Dolbeault cohomology. (PDF)

Lecture 4: Geometry of V+V*, linear Dirac structures, and generalized matrices. (PDF)

Lecture 5: Spinors, the spin group, a bilinear pairing on spinors, and pure spinors. (PDF)

Lecture 6: Generalized Hodge star, and spinors for TM+T*M and the Courant algebroid. (PDF)

Lecture 7: Exact Courant algebroids, and Severa’s classification of exact Courant algebroids. (PDF)

Lecture 8: Dirac structures, and geometry of Lie groups. (PDF)

Lecture 9: Bilinear forms on groups. (PDF)

Lecture 10: Integrability, Dirac maps, and manifolds with Courant structure. (PDF)

Lecture 11: Integrability and spinors, and Lie bialgebroids and deformations. (PDF)

Lecture 12-17: Generalized complex structures and topological obstructions, intermediate cases, spinorial description, and introduction to Hermitian geometry. (PDF)

Lecture 18: Generalized Kahler geometry. (PDF)

Lecture 19: Generalized Kahler geometry, and Hodge theory on generalized Kahler manifolds. (PDF)

Lecture 20: Generalized complex branes of rank 1. (PDF)

Lecture 21-23: Linear algebra, and T-duality. (PDF)

Course Info

As Taught In
Fall 2006
Learning Resource Types
Lecture Notes
Problem Sets